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Chapter 7: Q38E (page 346)

Consider a rotationT(x鈫�)=Ax鈫�in鈩�3in. (That is, A is an orthogonal 3x3matrix with determinant 1.) Show that T has a non-zero fixed point [i.e., a vectorT(x鈫�)=Ax鈫�in鈩�3withT(v鈫�)=v鈫�]. This result is known as Euler鈥檚 theorem, after the great Swiss mathematician Leonhard Euler (1707鈥�1783). Hint: Consider the characteristic polynomialrole="math" localid="1659595800447" fA(位). Pay attention to the intercepts with both axes. Use Theorem 7.1.4.

Short Answer

Expert verified

A is orthogonal, it applies .位=1.S0,鈭�x鈭�鈩�3,Ax=x,

Step by step solution

01

Theorem

The possible real eigenvalues of an orthogonal matrix are 1 and 鈭�1.

02

Solution of the problem

We have 0A=detA-0l=detA=1andim位鈫�鈭�,and , which means that 鈭�位鈭�0,鈭�,fA位=0. Since is orthogonal, it applies 位=1So,鈭�x鈭�鈩�3,Ax=x,

Therefore, the final result of is orthogonal, it applies So 位=1So,鈭�x鈭�鈩�3,Ax=x,

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Most popular questions from this chapter

Two interacting populations of coyotes and roadrunners can be modeled by the recursive equations

h(t + 1) = 4h(t)-2f(t)

f(t + 1) = h(t) + f(t).

For each of the initial populations given in parts (a) through (c), find closed formulas for h(t) and f(t).

(a)h(0)=f(0)=100(b)h(0)200,f(0)=100(c)h(0)600,f(0)=500

Question: If a vector v鈫�is an eigenvector of both An脳n, Awith associated eigenvalue 位, what can you say aboutker(A-位滨n)?Is the matrixA-位滨ninvertible?

For which2脳2 matrices A does there exist a nonzero matrix M Such that AM=MD,whereD=[2003] Give your answer in terms of eigenvalues of A.

Question: If a vectorv鈫�is an eigenvector of both AandB, isv鈫� necessarily an eigenvector ofAB?

26: Based on your answers in Exercises 24 and 25, sketch a phase portrait of the dynamical system

x炉(t+1)=[0.50.250.50.75]x炉(t)
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